BNL-96631-2011-CP

QCD Spin Physics: Partonic Spin Structure of the Nucleon

W. Vogelsang et al. Presented at the International School of Nuclear Physics; 33rd Course, From Quarks and

Gluons to Hadrons and Nuclei Erice-Sicily, Italy

September 16 to 24, 2011

December 2011

Physics Department/Nuclear Theory Group/Office of Science

Brookhaven National Laboratory

U.S. Department of Energy Office of Science, Nuclear Physics

Notice: This manuscript has been authored by employees of Brookhaven Science Associates, LLC under Contract No. DE-AC02-98CH10886 with the U.S. Department of Energy. The publisher by accepting the manuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. This preprint is intended for publication in a journal or proceedings. Since changes may be made before publication, it may not be cited or reproduced without the author’s permission.

DISCLAIMER

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QCD Spin Physics: Partonic Spin Structure of the Nucleon∗

D. de Florian,1 R. Sassot,1 M. Stratmann,2 W. Vogelsang3

1Departamento de Fisica, Universidad de Buenos Aires,Ciudad Universitaria, Pabellon 1 (1428) Buenos Aires, Argentina

2Physics Department, Brookhaven National Laboratory,Upton, NY 11973, U.S.A.

3Institute for Theoretical Physics, Universitat Tubingen,Auf der Morgenstelle 14, D-72076 Tubingen, Germany

December 6, 2011

Abstract

We discuss some recent developments concerning the nucleon’s helicity parton distribution func-tions: New preliminary data from jet production at RHIC suggest for the first time a non-vanishingpolarization of gluons in the nucleon. SIDIS measurements at COMPASS provide better constraintson the strange and light sea quark helicity distributions. Single-longitudinal spin asymmetries inW -boson production have been observed at RHIC and will ultimately give new insights into thelight quark and anti-quark helicity structure of the nucleon.

1 Introduction

QCD spin physics has been driven by the hugely successful experimental program of polarized deeply-inelastic lepton-nucleon scattering (DIS) [1]. One of the most important results has been the findingthat the quark and anti-quark spins (summed over all flavors) provide only about a quarter of thenucleon’s spin, ∆Σ ≈ 0.25 in the proton helicity sum rule [2]:

1

2=

1

2∆Σ+∆G+ Lq + Lg . (1)

This result implies that sizable contributions to the nucleon spin should come from the gluon spincontribution∆G, or from orbital angular momenta Lq,g of partons. To determine the other contributionsto the nucleon spin has become a key focus of the field. In the present article, we describe some ofthe recent developments of the field. We focus on current efforts to determine the helicity partondistributions of the nucleon and on the latest experimental results.

The helicity structure of the nucleon is foremost described by its twist-two helicity parton distributionfunctions,

∆f(x,Q2) ≡ f+(x,Q2) − f−(x,Q2) (f = u, d, s, u, d, s, g) , (2)

∗Talk presented by W. Vogelsang

1

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112.

0904

v1 [

hep-

ph]

5 D

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011

f+ (f−) denoting the number density of partons with the same (opposite) helicity as the nucleon’s, as afunction of momentum fraction x and scale Q. QCD predicts the Q2-dependence of the densities throughthe spin-dependent Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations [3]:

d

d lnQ2

�∆q∆g

�(x,Q2) =

�∆Pqq(αs, x) ∆Pqg(αs, x)∆Pgq(αs, x) ∆Pgg(αs, x)

�⊗

�∆q∆g

��x,Q2

�, (3)

where ⊗ denotes a convolution, and the splitting functions ∆Pij are evaluated in QCD perturbationtheory [3, 4, 5].

The contributions ∆Σ(Q2) and ∆G(Q2) in the helicity sum rule (1) are given by

∆Σ(Q2) =

� 1

0

�∆u+∆u+∆d+∆d+∆s+∆s

�(x,Q2)dx ≡ ∆Σu +∆Σd +∆Σs , (4)

∆G(Q2) =

� 1

0

∆g(x,Q2)dx . (5)

∆Σ is independent of Q2 at the lowest order. The distributions have a proper field-theoretic definition.For example, in case of ∆g it is given by [6]

∆g(x,Q2) =i

4π xP+

�dλ eiλxP

+ �P, S|G+ν(0) G+ν(λn)|P, S�

���Q2

, (6)

written in A+ = 0 gauge. Here, Gµν is the QCD field strength tensor, and Gµν its dual. The integralof ∆g(x,Q2) over all momentum fractions x becomes a local operator only in A+ = 0 gauge and thencoincides with ∆G(Q2) [2].

The helicity parton distributions may be probed in spin asymmetries for reactions at large momen-tum transfer. The probes used so far are inclusive and semi-inclusive deep-inelastic lepton scattering(DIS and SIDIS, respectively), and pp scattering at large transverse momentum, see Fig. 1. PolarizedDIS and SIDIS experiments have been carried out at SLAC, CERN, DESY and the Jefferson Laboratory[1] and mostly constrain the quark and anti-quark helicity distributions. RHIC at BNL [7, 8] is thefirst polarized proton-proton collider, operating at

√s = 200 and 500 GeV. The measurement of gluon

polarization in the proton is a major focus and strength of RHIC.The basic theoretical concept that underlies much of spin physics is the factorization theorem. It

states that large momentum-transfer reactions may be factorized into long-distance pieces that contain

!"#$ %%$&'(")*$#"!"#$

Figure 1: Parton-model Feynman diagrams for the processes constraining nucleon helicity structure.

2

the desired information on the spin structure of the nucleon in terms of its universal parton densities,and parts that are short-distance and describe the hard interactions of the partons. The latter can beevaluated using perturbation theory, thanks to the asymptotic freedom of QCD. As an example, weconsider the reaction pp → πX, where the pion is produced at high transverse momentum pT , ensuringlarge momentum transfer. The statement of the factorization theorem [9] is then:

d∆σ =�

a,b,c

∆fa ⊗ ∆fb ⊗ d∆σcab ⊗ Dπ

c (7)

for the polarized cross section, where ⊗ denotes a convolution. The Dπc are the pion fragmentation

functions. The sum in Eq. (7) is over all contributing partonic channels a + b → c + X, with d∆σcab

the associated spin-dependent partonic cross section. Factorization is valid up to corrections that aresuppressed as inverse powers of the hard scale. In general, a leading-order estimate of (7) merely capturesthe main features, but does not usually provide a quantitative understanding. Only with knowledge ofthe next-to-leading order (NLO) QCD corrections to the d∆σc

ab can one reliably extract information onthe parton distribution functions from the reaction. By now, NLO corrections are available for most ofthe processes relevant in polarized high-energy scattering [10].

Independent information on the nucleon’s helicity distributions may be obtained by using SU(2) andSU(3) flavor symmetries. The integrals of the flavor non-singlet combinations turn out to be proportionalto the nucleon matrix elements of the quark non-singlet axial currents, �P, S | q γµ γ5 λi q |P, S�. Suchcurrents typically occur in weak interactions, and one may relate the matrix elements to the β-decayparameters F,D of the baryon octet. One finds

∆Σu −∆Σd = F +D = 1.267,

∆Σu +∆Σd − 2∆Σs = 3F −D ≈ 0.58 . (8)

If valid, the second relation when combined with Eq. (4) gives that ∆Σ = 0.58+ 3∆Σs, so that a smallquark spin contribution to the proton spin implies a large negative strange quark contribution. Fairlysignificant violations of SU(3) symmetry have been predicted based on heavy baryon chiral perturbationtheory [11]. Lattice investigations of this issue have begun but are not yet conclusive [12].

2 Nucleon helicity structure: status 2009

In recent publications [13], we have presented the first next-to-leading order (NLO) “global” QCDanalysis of the nucleon’s helicity distribution from DIS, semi-inclusive DIS (SIDIS), and pp scatteringat RHIC. We have used a Mellin moment method for the analysis. Our results are shown in Fig. 2,along with estimates of their uncertainties. The shaded bands in Fig. 2 show the distributions that areallowed if one permits an overall increase of ∆χ2 = 1 (green) or ∆χ2/χ2 = 2% (yellow). As one cansee, the “total” ∆u+∆u and ∆d+∆d helicity distributions are very well constrained. This is expectedsince these distributions are primarily determined by the large body of inclusive DIS data. Our resultsagree well with the distributions obtained in previous and other recent analyses [14, 15, 16, 17, 18]which considered only the lepton scattering data.

The sea anti-quark distributions still carry rather large uncertainties, even though they are betterconstrained now than in previous analyses, thanks to the advent of more precise SIDIS data and of anew set of fragmentation functions [19] that describes the observables well in the unpolarized case. Wefind that the sea appears not to be SU(2)-flavor symmetric: the ∆u distribution is mainly positive, whilethe ∆d anti-quarks carry opposite polarization. This pattern has been predicted at least qualitativelyby a number of models [14, 20]. Already based on the Pauli principle one would expect that if valence-uquarks primarily spin along the proton spin direction, uu pairs in the sea will tend to have the u quark

3

Figure 2: Present status of the nucleon’s NLO helicity distributions according to the global analysisof Ref. [13]. The solid center lines show the best-fit result. The shaded bands provide uncertaintyestimates, using a criterion of ∆χ2 = 1 (inner bands) or ∆χ2/χ2 = 2% (outer bands) as allowedtolerance on the χ2 value of the fit. Also shown are results from earlier analyses [14, 15] of nucleon spinstructure from lepton scattering data alone.

polarized opposite to the proton. Hence, if such pairs are in a spin singlet, one expects ∆u > 0 and,by the same reasoning, ∆d < 0. We note that the uncertainties in SIDIS are still quite large, and it isin particular difficult to quantify the systematic uncertainty of the results related to the fragmentationmechanism at the relatively modest energies available.

The strange sea quark density shows a sign change. At moderately large x ∼ 0.1, it is constrainedby the SIDIS data, which prefer a positive ∆s. On the other hand, the inclusive DIS data combinedwith the constraints from baryon β-decays demand a negative integral of ∆s. As a consequence, ∆sobtains its negative integral purely from the contribution from low-x. Interestingly, there are initiallattice determinations of the integral ∆Σs [12], which point to small values. It is clearly important tounderstand the strange contribution to nucleon spin structure better.

Constraints on the spin-dependent gluon distribution ∆g predominantly come from RHIC. As canbe seen from Fig. 2, the gluon distribution turns out to be small in the region of momentum fraction,0.05 � x � 0.2, accessible at RHIC, quite possibly having a node. At Q2 = 10 GeV2, the integral overthe mostly probed x-region is found to be almost zero,

� 0.2

0.05 dx∆g(x) = 0.005± 0.06, where the error isobtained for a variation of χ2 by one unit. Thus, on the basis of [13], there are no indications of a sizablecontribution of gluon spins to the proton spin. We also note that a way to access ∆g in lepton-nucleonscattering at HERMES and COMPASS is to measure final states that select the photon-gluon fusionprocess, heavy-flavor production and high-pT hadron or hadron-pair production [21, 22]. These datawere not included in the analysis [13], mostly because of the fact that success of the perturbative-QCDhard-scattering description had not been established for these observables in the kinematic regime of

4

0

0.2

0.4

0.6

0.8

-0.2

0

0.2

0.4

0.6

10 -2 10 -1

COMPASS Ap,π+COMPASS A1 COMPASS Ap,K+COMPASS A1

COMPASS Ap,π-COMPASS A1

x

DSSVDSSV+

COMPASS Ap,K-COMPASS A1

x10 -2 10 -1 1

Figure 3: COMPASS results [27] for SIDIS spin asymmetries on a proton target, compared to DSSV [13]and DSSV+ fits [29].

interest here. At least for single-inclusive high-pT hadrons in γp → h±X it has now been found thatQCD hard-scattering does appear to be applicable in the COMPASS kinematic regime [23, 24].

3 Recent developments

Interesting new developments have taken place following the original DSSV analysis, mostly related tothe advent of new data. We will summarize these in the following.

3.1 Recent DIS and SIDIS data

Recently, the COMPASS collaboration has published new DIS [25] and SIDIS [26, 27] data. The latterextend the coverage in x down to about x � 5 × 10−3, almost an order of magnitude lower than thekinematic reach of the HERMES data [28] used in the DSSV global analysis of 2008 [13]. For the firsttime, the new results comprise measurements of identified pions and kaons taken with a longitudinallypolarized proton target. Clearly, these data can have a significant impact on fits of helicity PDFs andestimates of their uncertainties. In particular, the new kaon data are expected to serve as an importantcheck of the validity of the strangeness density obtained in the DSSV analysis discussed above, whichinstead of favoring a negative polarization as in most fits based exclusively on DIS data, prefers avanishing or perhaps even slightly positive ∆s in the measured range of x.

Figure 3 shows a detailed comparison [29] between the new proton SIDIS spin asymmetries fromCOMPASS [26, 27] and the original DSSV fit (dashed lines). Also shown is the result of a re-analysisat NLO [29] accuracy (denoted as “DSSV+”) based on the updated data set. The differences betweenthe DSSV and the DSSV+ fits are hard to notice, both for identified pions and kaons. The total χ2

5

Figure 4: χ2 profiles for the truncated first moment of ∆s in two different x intervals, 0.02 ≤ x ≤ 1(left) and 0.001 ≤ x ≤ 0.02 (right).

of the fit drops only by a few units upon refitting, which is not really a significant improvement for aPDF analysis in view of non-Gaussian theoretical uncertainties. The change in χ2 is also well withinthe maximum ∆χ2/χ2 = 2% criterion adopted in the original DSSV global analysis [13]. Overall, uponrefitting, one finds a trend towards smaller net polarization for ∆u and ∆d in the range 0.001 ≤ x ≤ 1than in DSSV.

As we saw earlier, the original DSSV fit [13] found an interesting feature for the strangeness helicitydistribution: ∆s was found to be small and slightly positive at medium-to-large x, but has a significantlynegative first moment in accordance with expectations based on SU(3) symmetry and fits to DIS dataonly. To investigate this issue further, we present in Fig. 4 the χ2 profiles for two different intervalsin x: 0.02 ≤ x ≤ 1 (left) and 0.001 ≤ x ≤ 0.02 (right). The profiles in Fig. 4 clearly show that theresult for ∆s for 0.001 ≤ x ≤ 0.02 is a compromise between DIS and SIDIS data, the latter favoringless negative values. Interestingly though, the new COMPASS SIDIS data, which extend towards thesmallest x values so far, actually show some preference for a slightly negative value for ∆s as well.For 0.02 ≤ x ≤ 1 everything is determined by SIDIS data, and all sets consistently ask for a small,slightly positive strange quark polarization. There is no hint of a tension with DIS data here as theydo not provide a useful constraint at medium-to-large x. We note that at low x most SIDIS sets in theoriginal DSSV fit give indifferent results. We also mention that in the range x > 0.001 the hyperondecay constants, the so-called F and D values, do not play a significant role in constraining ∆s(x).To quantify possible SU(3) breaking effects one needs to probe ∆s(x) at even smaller values of x, forinstance in SIDIS at a future EIC [30]. We finally note that the HERMES and COMPASS data areconsistent in the region of overlap, 0.02 ≤ x ≤ 1.

Clearly, all current extractions of ∆s from SIDIS data suffer from a significant dependence on kaonFFs, see, e.g., Refs. [26, 27], and better determinations of DK(z) are highly desirable. Contrary to otherfits of FFs [32], only the DSS sets [19] provide a satisfactory description of pion and kaon multiplicitiesin the same kinematic range where we have polarized SIDIS data.

6

3.2 W bosons at RHIC

We have seen in the previous section that the SIDIS data provide some insights into the flavor structureof the polarized sea distributions of the nucleon, albeit with still fairly large uncertainties. Complemen-tary and clean information on ∆u, ∆u, ∆d, and ∆d will come from pp → W±X at RHIC, where oneexploits the maximally parity-violating couplings of the produced W bosons to left-handed quarks andright-handed anti-quarks [7, 33]. These give rise to a single-longitudinal spin asymmetry,

AL ≡ σ+ − σ−

σ+ + σ− , (9)

for the processes �pp → �±X, where the arrow denotes a longitudinally polarized proton and � = e orµ is the charged decay lepton. The high scale set by the W boson mass makes it possible to extractquark and anti-quark polarizations from inclusive lepton single-spin asymmetries inW boson productionwith minimal theoretical uncertainties, as higher order and sub-leading terms in the perturbative QCDexpansion are suppressed [34, 35, 36, 37].

For W− production, neglecting all partonic processes but the dominant ud → W− one, the spin-dependent cross section in the numerator of the asymmetry is found to be proportional to the combi-nation

∆σ ∝ ∆u(x1)⊗ d(x2)(1− cos θ)2 −∆d(x1)⊗ u(x2)(1 + cos θ)2 , (10)

where θ is the polar angle of the electron in the partonic c.m.s., with θ = 0 in the forward direction ofthe polarized parton. At large negative pseudorapidity ηlept of the charged lepton, one has x2 � x1 andθ � π/2. In this case, the first term in Eq. (10) strongly dominates, since the combination of partondistributions, ∆u(x1)d(x2), and the angular factor, (1 − cos θ)2, each dominate over their counterpartin the second term. Since the denominator of AL is proportional to u(x1)⊗ d(x2)(1− cos θ)2 + d(x1)⊗u(x2)(1 + cos θ)2, the asymmetry provides a clean probe of ∆u(x1)/u(x1) at medium values of x1. Bysimilar reasoning, at forward rapidity ηlept � 0 the second term in Eq. (10) dominates, giving access to−∆d(x1)/d(x1) at relatively high x1.

For W+ production, one has the following structure of the spin-dependent cross section:

∆σ ∝ ∆d(x1)⊗ u(x2)(1 + cos θ)2 −∆u(x1)⊗ d(x2)(1− cos θ)2 . (11)

Here the distinction of the two contributions by considering large negative or positive lepton rapidities isless clear-cut than in the case ofW−. For example, at negative ηlept the partonic combination d(x1)u(x2)will dominate, but at the same time θ � π/2 so that the angular factor (1 + cos θ)2 is small. Likewise,at positive ηlept the dominant partonic combination ∆u(x1)d(x2) is suppressed by the angular factor.So both terms in Eq. (11) will compete essentially for all ηlept of interest. Our global analysis techniqueis of course suited for extracting information on the polarized PDFs even if there is no single dominantpartonic subprocess. The NLO corrections to the single-inclusive lepton cross sections have recentlybeen presented in [37] in a way tailored to use in the global analysis framework.

Figure 5 shows the first published data from RHIC for AL in W± production [38, 39]. For now, thestatistical uncertainties are still large. However, already now a large negative asymmetry is seen for thecase of W+ production, resulting primarily from the positive up-quark polarization in the proton (seeEq. (11)). Clearly, there is a large potential in future W -measurements at RHIC.

3.3 New constraints on ∆g

The STAR and PHENIX experiments at RHIC have recently presented new preliminary data fromthe 2009 run for the double-helicity spin asymmetry ALL for jet and neutral-pion production, respec-tively [40, 41]. The results are shown in Figs. 6 and 7. One can see that the experimental uncertainties

7

e2 1 0 1 2

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

RHICBOS +W W

K-DNS KKP-DNS 08DSSV

08DSSV CHE

Syst. uncertainty due to background, w/o pol. norm. uncertainty of 9.2%

V500 Ge = s STAR

X + ±e X + ±W p + p

V < 50 Gee TE 25 <

W

+W

LA

+ +

+ = LA

Figure 5: Published STAR [38] (top) and PHENIX [39] (bottom) data for the single-helicity asymmetryAL in W± production at RHIC.

are very significantly reduced as compared to those in the previous run-6 data sets [42, 43]. An inter-esting feature of the new preliminary STAR data is that they lie consistently above the result for thebest-fit DSSV distribution for jet transverse momenta below 25 GeV or so. They do remain well belowthe old GRSV-“standard” result of [14], on the other hand. This suggests that the spin-dependentgluon distribution may be somewhat different from zero in the x-range where it is constrained by theRHIC data. The thick (magenta) solid line in the figure shows ALL obtained for a special set of par-ton distributions within the DSSV analysis. For this set the truncated moment of ∆g over the region0.001 ≤ x ≤ 1 was varied, allowing the total χ2 to change by 2%. Evidently, this set of parton distribu-tions describes the STAR data rather well. The truncated moment of ∆g in this set over the x-rangeaccessed at RHIC is

� 0.2

0.05

dx∆g(x,Q2 = 10 GeV2) = 0.13 , (12)

8

Figure 6: Preliminary STAR run-9 data [40] for the double-helicity asymmetry ALL for single-inclusivejet production, as a function of jet transverse momentum for |η| < 1. The preliminary run-6 data areshown as well. The theoretical curves are as described in the caption. The additional solid magentaline gives the result for a special DSSV set of polarized parton distributions, for which the truncatedmoment of ∆g over the region 0.001 ≤ x ≤ 1 was varied allowing the total χ2 of the fit to change by2%.

which is just within the range� 0.2

0.05

dx∆g(x,Q2 = 10 GeV2) = 0.005+0.129−0.164 (13)

quoted as more conservative uncertainty (∆χ2/χ2 = 2%) in [13]. Despite the fact that this really isonly an illustration that cannot replace a proper re-analysis of the data, it does appears that, for thefirst time, there are indications of non-vanishing gluon polarization in the nucleon. Figure 7 shows thecomparison to the new preliminary PHENIX π0 data [41]. Here the values of ALL are much smaller,which is mostly due to the fact that lower values of x are probed at the transverse momenta relevant inthe PHENIX measurements. One can see that the data are well described by both the DSSV set andthe special set of polarized parton distributions used in Fig. 6.

4 The future: Electron Ion Collider (EIC)

An Electron Ion Collider is currently being considered in the U.S. as a new frontier facility to explorestrong-interaction phenomena [31]. One of its key features would be the availability of high-energy,high-luminosity polarized ep collisions to probe nucleon spin structure. This would also allow precisionextractions of ∆g, in particular from scaling violations of the proton’s spin-dependent structure functiong1. Figure 8 shows the results of a recent dedicated phenomenological study [44]. “Pseudo” EIC-datawere generated for collisions of 5 GeV electrons with 50, 100, 250, and 325 GeV protons and were addedto the DSSV global analysis code. The statistical precision of the data sets for 100− 325 GeV protonswas taken to correspond to about two months of running at the anticipated luminosities for eRHICwith an assumed operations efficiency of 50%. For 5 × 50GeV an integrated luminosity of 5 fb−1 was

9

!""#$

Figure 7: As in Fig. 6, but for the preliminary PHENIX run-9 data [41].

assumed. The projected uncertainties were used to randomize the pseudo-data by one sigma aroundtheir central values determined by the DSSV set of PDFs. With these projected EIC data with theirestimated uncertainties, a re-fit of the DSSV polarized parton distribution functions was performed.The results are shown in Fig. 8. As one can see, with EIC data it should be possible to map thecurrently completely undetermined shape of ∆g for 10−4 � x � 0.01 to an accuracy of about ±10% orbetter.

Acknowledgments

This work was supported in part by the U.S. Department of Energy (contract number DE-AC02-98CH10886), and by CONICET, ANPCyT and UBACyT.

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